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This issue is particularly apparent in computational mathematics. Traditional models of computation such as the Turing machine have storage capacity which is unbounded, so it is in principle possible to perform computations with the elements of infinite sets.
On the other hand, since the amount of storage space in use at any one time is finite, we need each element to have a finite representation. For various reasons, it is not always possible or desirable to use a system of unique encodings, that is, one in which every element has a single encoding.
When using an encoding system without uniqueness, the question naturally arises of whether there is an algorithm which, given as input two encodings, decides whether they represent the same element.
Such an algorithm is called a solution to the word problem for the encoding system. The word problem in combinatorial calculus[ edit ] Main article: Because combinators encode all possible Turing machinesand the equivalence of two Turing machines is undecidable, it follows that the equivalence of two strings of combinators is undecidable.
Likewise, one has essentially the same problem in untyped lambda calculus: For several typed variants of the lambda calculus, equivalence is decidable by comparison of normal forms.
The word problem in universal algebra[ edit ] In algebraone often studies infinite algebras which are generated under the finitary operations of the algebra by finitely many elements. In this case, the elements of the algebra have a natural system of finite encoding as expressions in terms of the generators and operations.
The word problem here is thus to determine, given two such expressions, whether they represent the same element of the algebra. Roughly speaking, the word problem in an algebra is: While the former asks whether two terms are equal, the latter asks whether they have instances that are equal.
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we cannot assume that any writer shares our exact perspective on all points. Right Word, Wrong Meaning We may even read a word, know the meaning of it, and still misunderstand the meaning the a given passage and a careful rewriting.
We might think of paraphrasing as a trick that forces us To paraphrase is to restate a passage precisely. We employed a lawyer to do the rewrite, but by doing away with reprinting an annual rule book, the cost of rewriting will be cost-neutral in the long term.
BHA rewords rules to make them clearer That's the first lesson behind the "don't rewrite from scratch" rule. All forms discussed are shown to be of relevance for word and music studies.
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